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Hausdorff dimension of affine random covering sets in torus

Esa JärvenpääMaarit JärvenpääHenna KoivusaloBing LiVille Suomala — 2014

Annales de l'I.H.P. Probabilités et statistiques

We calculate the almost sure Hausdorff dimension of the random covering set lim sup n ( g n + ξ n ) in d -dimensional torus 𝕋 d , where the sets g n 𝕋 d are parallelepipeds, or more generally, linear images of a set with nonempty interior, and ξ n 𝕋 d are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.

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